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Retarded van der Waals Forces between Triblock-Coated Latex Spheres X. Wu and T. G. M. van de Ven* Paprican and Department of Chemistry, Pulp and Paper Research Centre, McGill University, Montreal, Quebec, Canada H3A 2A7 Received June 14, 1996. In Final Form: September 18, 1996X Retarded van der Waals forces between colloidal particles play a major role in colloid and surface science, but no direct measurement of these forces has been reported. A new experimental technique has been developed to accurately determine these forces between two 4.7 µm latex spheres coated by triblock copolymers. The retardation wavelength has been found to be 110 ( 10 nm and the Hamaker constant (1.0 ( 0.1) × 10-20 J. Both of them agree well with theoretical predictions. In addition, the experiments allow the measurement of the layer thickness of adsorbed triblock copolymers. The results are consistent with the data from field-flow fractionation and photon correlation spectroscopy.
Introduction Van der Waals forces are of great interest in colloid and surface science because they play an important role in many physical and chemical phenomena, such as colloid stability, physical adsorption, adhesion, surface tension, wetting, foam stability, strength of clay minerals, and micellar structure.1,2 The forces are known to be retarded at large separation distances (about 5 nm for two molecules in free space),2 because of the finite time it takes for an electromagnetic signal to travel forth and back between two molecules. This retardation effect was first studied by Casimir and Polder3 and later implicitly incorporated in the full Lifshitz treatment.4 However, adequate data for accurate Lifshitz calculations are lacking except for a few systems; so for practical purposes the Casimir and Polder approach is usually applied to calculate retarded van der Waals forces.5 Another approach, widely used because of its simplicity, was introduced by Hamaker.2 Unlike the Lifshitz treatment, it does not take retardation into account, but modifications have been carried out to include this effect6,7 which yield similar results as exact calculations.5,7 Since the van der Waals forces that determine colloid stability are almost always retarded, it is of interest to measure these forces between colloidal particles experimentally. The commonly used surface force apparatus (SFA)8 or atomic force microscope (AFM)9 are unfortunately not sensitive enough to detect these colloidal forces, although the retarded van der Waals forces between two macroscopic mica surfaces in aqueous solutions, which are 5 orders of magnitude larger than the colloidal forces, were measured with SFA by Israelachvili et al.8 Recently a new method, colloidal particle scattering (CPS), was developed by us to measure surface * Corresponding author: tel, 514-398-6177; fax, 514-398-6256; e-mail,
[email protected]. X Abstract published in Advance ACS Abstracts, December 1, 1996. (1) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1989; Vol. 1, p 620. (2) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1985; p 152. (3) Casimir, H. B. G.; Polder, D. Phys. Rev. 1948, 73, 360. (4) Lifshitz, E. M. Sov. Phys. JETP 1956, 2, 73. (5) Clayfield, E. J.; Lumb, E. C.; Mackey, P. H. J. Colloid Interface Sci. 1971, 37, 382. (6) Schenkel, J. H.; Kitchener, J. A. Trans. Faraday Soc. 1960, 56, 161. (7) Gregory, J. J. Colloid Interface Sci. 1981, 83, 138. (8) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (9) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239.
S0743-7463(96)00589-6 CCC: $12.00
Figure 1. Schematics of two colliding latex particles. One particle is attached to a glass surface, while the other one moves in a wall shear flow. The motion is created by moving the bottom of the cell containing latex particles parallel to a fixed top plate to which the immobilized sphere sticks. G is the shear rate and V is the flow velocity. Ls is the thickness of the adsorption layer on the latex surface, a is the radius of the particle, and h is the separation gap width used in hydrodynamic calculations. In the van der Waals force equation, the distance between the two surfaces is taken as hvdw ) h + 2Ls. The dotted line shows the trajectory of the mobile particle. The trajectory of approach is characterized by two coordinates: x and z prior to the hydrodynamic and colloidal interactions, while the trajectory of recession is characterized by another set of x and z after the collision.
forces.10 The forces detected by CPS are at least 4 orders of magnitude smaller than those detected by SFA or AFM. In this paper we determine the retarded van der Waals forces between latex spheres coated with triblock copolymers which are widely used as stabilizers for colloidal particles. As an additional advantage, the thickness of adsorbed polymer layers on latexes can also be determined. Few other methods are available to measure such thicknesses for large latex particles, since neither dynamic light scattering nor scanning electron microscopy (SEM) techniques are suitable, because of the small decrease in the diffusion constant and the collapse of the polymer layer, respectively. Theory The CPS method is based on the analysis of particle collision trajectories between an immobile particle stuck to a wall and a freely mobile particle subjected to a wall shear flow of known shear rate (cf. Figure 1). Usually collision trajectories of approach are different from those of recession, and thus the mobile particle is “scattered” by (10) van de Ven, T. G. M; Warszynski, P.; Wu, X.; Dabros, T. Langmuir 1994, 10, 3046.
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the immobile particle, similar as in elementary particle scattering. Initial positions of mobile spheres can be controlled by a micromanipulator, and collisions can be generated at will at a rate of several hundred per day. These collision trajectories are recorded and analyzed with an image processing board. The fitting of the theoretically calculated trajectories to the experimentally observed ones, by varying the parameters in the colloidal force equations, yields the optimum parameters which then gives the force-distance profile. The technique is discussed in detail in ref 10, in which we measured the surface potential and Hamaker constant of polystyrene latex particles assuming the force-distance profile follows the classical DLVO theory of colloid stability. The results agree with zeta-potential measurements and the theoretical prediction of the Hamaker constant between polystyrene and polystyrene in water. Since then the data processing technique has been improved to yield even more reliable results and the original smooth latex model has been replaced with a more realistic hairy latex model11 to account for various observations with dynamic light scattering12 and electrophoresis.13 A similar model can be applied to triblock-coated latex spheres. The polymer is assumed to be evenly coated on the latex surface and form a layer of thickness Ls (cf. Figure 1). When the particle moves in water, we assume that the water trapped in the adlayer is immobile, so the particle radius in hydrodynamic calculations is taken as a + Ls instead of a. Since the adlayer is mostly composed of water, the triblocks contribute very little to the van der Waals force. In our model this contribution is neglected completely and the gap width in the equation of the van der Waals force is taken as h + 2Ls. This model is valid only when two adlayers do not overlap during a collision. For our system we know from literature data14,15 that the adlayer thickness is about 10 nm. If we control the salt (KCl) concentration at 0.01 M, the minimum gap width between two particle surfaces during a collision (about 30 nm) will be larger than the sum of the two adlayers. This ensures that the laryers do not overlap. According to this model, no steric repulsive force is involved and the classical DLVO theory which considers two colloidal forces, i.e., electrostatic and van der Waals forces, describes the system reasonably well. Two parameters enter in the electrostatic force: the Debye length, κ-1, and the surface potential, ψ0; κ-1 can be accurately calculated from the known salt concentration, while ψ0 is not very sensitive to collision trajectories at the experimental conditions (0.01 M KCl) and can be kept constant at the measured zeta-potential. Hence the parameters to be fitted come solely from the retarded van der Waals force, and the accuracy of this force measurement is maximized. Because the total number of parameters to be fitted is limited (usually two), we need simple expressions for the van der Waals force. The Hamaker expression modified by Schenkel and Kitchener’s retardation term6 is adequate in this case. It is determined by two parameters: the Hamaker constant, A, and the retardation wavelength, λ, usually of the order of 100 nm.2,6,16 It further depends on the geometry of the colliding particles, i.e., the particle radius a (a constant value given by the manufacturer) and the adlayer thickness Ls. According (11) X. Wu and T. G. M. van de Ven Langmuir 1996, 12, 3859. (12) Polverari, M.; van de Ven, T. G. M. Colloids Surf., A: Physicochem. Eng. Aspects 1994, 86, 209. (13) Chow, R. S.; Takamura, K. J. Colloid Interface Sci. 1988, 125, 226. (14) Li, J.-T.; Caldwell, K. D.; Rapoport, N. Langmuir 1994, 10, 4475. (15) Baker, J. A.; Berg, J. C. Langmuir 1988, 4, 1055. (16) van de Ven, T. G. M. Colloidal Hydrodynamics; Academic Press: London, 1989; p 24.
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Figure 2. Scattering diagrams of 25 selected collisions from two Pluronic triblock-coated latex experiments: (a) F88-coated latex, zeta-potential -23 mV; (b) F108-coated latex, zetapotential -20 mV. The coordinates are scaled by the particle radius. The open circles represent the initial positions of the moving particle (one set of x and z), the filled circles stand for the experimentally observed final positions of the particle (another set of x and z), and the filled triangles are the theoretically calculated final positions using the best-fit parameters in the force equations. The dotted semicircle is the projection of the stationary particle.
to our previous results,10 the choice of λ ) 100 nm is a reasonable one. A more detailed discussion will be given below. A and Ls are thus the two parameters to be fitted by experimental data. Experimental Section The latex spheres used in the experiments are 4.7 µm surfactant-free polystyrene latexes supplied by Interfacial Dynamics Corp. The sample has undergone heat-treatment above the glass transition temperature, Tg, to smooth the surface.13 The polymers used for adsorption are Pluronic F88 and Pluronic F108 supplied by BASF Corp. They are PEOPPO-PEO [PEO being poly(ethylene oxide) and PPO being poly(propylene oxide)] triblock copolymers with different sizes of blocks (104/39/104 repeat units for F88 and 129/56/129 repeat units for F108). The coated samples were made by dropping the bare latex suspension into a 50 mg/L Pluronic F108 or F88 surfactant solution and leaving it overnight on a rotary machine to prevent sedimentation. The samples were then separated from the solutions by centrifuging and supernatant removal. Before each experiment the polymer-coated latex spheres were dispersed in a 0.01 M KCl solution consisting of 60% D2O and 40% H2O, a mixture in which the latex spheres are density matched. Both D2O and H2O were double-distilled to minimize contamination. In aqueous solution PPO is the anchor-block on the surface of latex particles due to its higher hydrophobicity and PEO is the buoy-block extending into solution.17 The adlayer thickness, Ls, is thus the thickness of this PEO layer. No desorption of PEO-PPO-PEO is expected when the coated spheres are suspended in the KCl solution, and comparisons of the results of this study with other techniques (see below) confirm this. In each experimental run, more than 100 collision trajectories were analyzed. From these collisions, 25 were selected according to a criterion discussed in ref 11, in essence rejecting collisions for which the tangential deviation is not small compared with (17) Lee, J.; Martic, P. A.; Tan, J. S. J. Colloid Interface Sci. 1989, 131, 252.
Forces between Colloidal Particles
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Figure 4. χ2 contour map of λ vs Ls. The Hamaker constant, A, is fixed at the theoretical value, 9.5 × 10-21 J. The minimum gives λ ) 110 ( 10 nm.
Figure 3. Contour maps of the least-squares fitting parameter χ2 for two Pluronic triblock-coated latex experiments: (a) F88coated latex, minimum at (Ls ) 7 nm, A ) 1.0 × 10-20 J); (b) F108-coated latex, minimum at (Ls ) 11 nm, A ) 1.0 × 10-20 J). The χ2 values of contour lines are indicated by the numbers in the figure. the radial displacement;11 this procedure minimizes the error caused by surface roughness. These selected collisions are plotted in a scattering diagram (cf. Figure 2) where the open circles represent the positions of the moving particle before the collision, the filled circles represent the positions after the collision, and the filled triangles are the theoretical final positions (to be discussed shortly). The x-coordinates can be obtained directly from the video images, while the z-coordinates are obtained from the measured particle velocities.10
Results and Discussion From these selected collisions we can obtain the van der Waals force by finding the values of A and Ls which give the best fit between observed and theoretical collision trajectories. The fitting is carried out by plotting a contour map of χ2, which indicates the goodness-of-fit,10 at different values of A and Ls (cf. Figure 3). The minimum value of χ2 on the map gives the optimum values of A and Ls. The trends in the contour lines in Figure 3 show the expected dependency, i.e. an increase in Ls requires an increase in A to compensate for the effect of a thicker adlayer. For this reason, most of the minima and contour lines are aligned diagonally from bottom left to up right. The global minima for F88- and F108-coated latexes are at (Ls ) 7 nm, A ) 1.0 × 10-20 J) and (Ls ) 11 nm, A ) 1.0 × 10-20 J), respectively (cf. Figure 3). The adlayer thicknesses of these two triblocks have been measured with field-flow fractionation (FFF).14 For F88 and F108 adsorbed on 272 nm latex particles the results are 9 and 14 nm. Despite the particle size difference
between 272 nm latex and our 4.7 µm IDC latex, the curvature effects on the adsorption are very small due to the relatively thin adlayers.18 Hence the results are comparable. The adsorption layer thicknesses determined by our method are slightly smaller than FFF results. This is due to the lower equilibrium polymer concentration of F88 and F108 (50 mg/L) in our experiments, which is at the beginning of the plateau in the adsorption isotherms.15,19 Maximum adsorption occurs at 200 mg/L, and the maximum adsorption layer thickness (13 nm for F108 adsorption) occurs at 500 mg/L measured by photon correlation spectroscopy (PCS) on 56 nm latex particles.15 At 50 mg/L polymer concentration, the layer thickness of F108 is determined to be 10 nm,15 which corresponds to about 11.5 nm for the same adlayer on a 4.7 µm sphere after correcting for curvature.18 This result agrees very well with our measurement (11 nm) at the same conditions. CPS actually has some advantages over both FFF and PCS. To obtain an accurate result of particle size with FFF, it is necessary to know the exact value of particle density (including the adlayer) which could be difficult to obtain when the particle is coated with a polymer layer whose thickness is to be determined. PCS is thus used frequently to calibrate FFF results.20,21 However, PCS requires “perfectly smooth” bare latex spheres, and the layer thickness is obtained by subtracting the bare radius from the polymer-coated sphere radius. The presence of a hairy layer on bare latex spheres thus causes a problem for PCS measurements. This limitation does not apply to CPS as long as the hairy layer of the uncoated particles is thinner than the adsorption layer. The coil diameters of the freely dissolved F88 and F108 Pluronic surfactants have been determined by Lee et al.17 They are 5.8 and 8.1 nm, respectively. Both of them are smaller than the measured adlayer thicknesses: 7 and 11 nm. This indicates that the buoy-block (PEO) is in the form of more extended “brushes” rather than coils or “mushrooms”. As predicted by de Gennes22 and Alexander,23 an increase in polymer adsorption or anchored (18) Garvey, M. J.; Tadros, Th. F.; Vincent, B. J. Colloid Interface Sci. 1976, 55, 440. (19) Kayes, J. B.; Rawlins, D. A. Colloid Polym. Sci. 1979, 257, 622. (20) Caldwell, K. D.; Jones, H. K.; Giddings, J. C. Colloids Surf. 1986, 18, 123. (21) Caldwell, K. D.; Li, J. J. Colloid Interface Sci. 1989, 132, 256. (22) de Gennes, P. G. Adv. Colloid Interface Sci. 1987, 27, 189. (23) Alexander, S. J. Phys. (Paris) 1977, 38, 983.
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(or grafted) chain density will change the configuration of the buoy-block from a “mushroom” to a “brush”. A further increase in adsorption up to the plateau region will make the “brushes” even more extended. This explains the 2-3 nm increase in the adsorption layer thicknesses of F88 and F108 triblocks when their equilibrium concentrations increase from 50 to 500 mg/L. The retardation wavelength λ is a measure of how and where the transition from nonretarded to retarded van der Waals forces occurs. All of the obtained results are based on the assumption that λ ) 100 nm. The validity of this assumption can be tested by fixing the Hamaker constant at the theoretical value, 9.5 × 10-21 J, calculated from spectroscopic data,1 and constructing a χ2 contour map of λ vs Ls. For F108-coated latexes the result is shown in Figure 4. The best fit yields λ ) 110 ( 10 nm, which is consistent with our previous assumption that λ ) 100 nm. After obtaining the optimum values of the parameters, we can plot the theoretical predictions of the particle’s final positions in scattering diagrams (cf. triangles in Figure 2). The discrepancy between experimental and theoretical positions is caused by surface roughness (of the two colliding particles) and Brownian motion (of the moving particle). According to our error analysis with a Monte Carlo method,11 these errors result in deviations of 1 nm and 1 × 10-21 J for Ls and A respectively. The force-distance profile can be plotted according to the DLVO theory once the force parameters are known (cf. Figure 5). The total colloidal interaction force consists of an electrostatic repulsion, dominant below about 30 nm, and a van der Waals attraction, dominant at large separation. Equations for these forces have been given in refs 10 and 11. The electrostatic force is mainly determined by the ionic strength (or equivalently the Debye length κ-1) and the zeta-potential of the spheres, while the van der Waals force is determined by the Hamacker constant A, the retardation wavelength λ, and the thickness Ls of the adsorbed triblock layer. Note that forces as small as 10 fN (10-14 N) can be readily determined. The profile with the nonretarded van der Waals force (λ f ∞) is also plotted for comparison, showing that retardaton weakens the force considerably.
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Figure 5. A fitted DLVO-type force with a retarded van der Waals force term vs separation gap width (solid curve) for two triblock-coated latex spheres. The force parameters are taken from the measurement of the Pluronic F108-coated latex experiment, i.e., Ls ) 11 nm, A ) 1.0 × 10-20 J, and λ ) 100 nm. The force unit is femtonewton (10-15 N). The interaction force with the same Ls and A but neglecting the retardation effect is also plotted (dotted curve) for comparison. Note the large effect of retardation on the force.
Conclusion We have shown that CPS is capable of determining accurately the retarded van der Waals force between two latex spheres, while at the same time providing the thickness of polymer adlayers. The results for Pluronic triblock-coated latexes are A ) (1.0 ( 0.1) × 10-20 J, λ ) 110 ( 10 nm, and Ls ) 7 and 11 nm ((1 nm) for F88 and F108, respectively. Because the model only applies when the adsorption layers do not overlap, it is limited to particles with thin adlayers. However, since most surfactant coatings are within 10 nm, it can actually be applied to a large number of systems. LA9605899
